abc and foundations


Now that Davide Castelvecchi’s lucid Nature article on Mochizuki’s “impenetrable” work on the abc Conjecture has been reprinted by Scientific American, many of us can expect our non-expert friends to ask us what’s going on.  (It has already happened to me.)  If your non-expert friends happen to be sociologists, please advise them to review Castelvecchi’s text for clues to consensus-formation within our own impenetrable community.  Castelvecchi observes that Mochizuki makes the parallel explicit:

Mochizuki wrote that the status of his theory with respect to arithmetic geometry “constitutes a sort of faithful miniature model of the status of pure mathematics in human society”.

I don’t even have to resist the temptation to offer my own opinion on the correctness of Mochizuki’s work, because I decided early on, after spending less than half an hour with the manuscripts, to wait for consensus-formation before thinking about the question.  As the Nature article reports, even experts strongly invested in the abc Conjecture, as I am not, remain stymied by Mochizuki’s approach.  I’m happy to let them sort it out, in Oxford in December or at another place and time.

Instead I want to focus on this passage from Castelvecchi’s article:

He is attempting to reform mathematics from the ground up, starting from its foundations in the theory of sets… And most mathematicians have been reluctant to invest the time necessary to understand the work because they see no clear reward…

(Sociologists, please note the word “reward.”)  Readers of this blog will be aware that Mochizuki’s is not the only attempt to “reform mathematics from the ground up” and that a lively and opinionated subculture is hoping to convince mathematicians to “invest the time necessary” not only to understand homotopy type theory but to consider adopting it as a new foundation for mathematics. It’s not clear from the lengthy discussion here and elsewhere on this blog whether HoTT has yet unveiled a “killer app” as capable of capturing the community’s imagination as the abc Conjecture undoubtedly would be, if Mochizuki’s proof turned out to be correct.  Castelvecchi (who is tactful enough not to repeat this rumor) leaves one point ambiguous, and I haven’t seen it clarified elsewhere (but I may simply not have been looking carefully enough):   would all the rest of number theory still be OK under Mochizuki’s proposed “reform”?  Can we imagine a situation in which we get to keep either the abc Conjecture OR Deligne’s purity theorem for l-adic cohomology, for example, but not both?  (And did we really just imagine that situation, or did we merely imagine that we were imagining it?)

Number theory appeared to be facing an equally stark but much less consequential dilemma nearly 20 years ago, when several mathematicians expressed concern that what appeared to be an established proof that the Langlands conjecture on conjugation of Shimura varieties (completely established by Borovoi and Milne) implied the existence of canonical models in the sense of Shimura and Deligne was based on a flawed interpretation of Weil’s theorem on Galois descent.  The issue seemed to be that there was no clear way to translate the version of Weil’s theorem implicitly used by Langlands, Milne, and others, which could be traced back to some of Shimura’s early work and was therefore formulated in the language of Weil’s Foundations of Algebraic Geometry, into Grothendieck’s theory of faithfully flat descent for schemes, which by that time was the only language anyone cared to use.   Some mathematicians went so far as to speculate that Shimura had got it wrong and that the whole theory of Shimura varieties was unreliable; others were ready to entertain  the unlikely but not altogether inconceivable possibility that Weil’s and Grothendieck’s foundations were actually incompatible.  In the end Milne found an updated version of what most specialists decided was probably the proof Shimura had in mind, and Yakov Varshavsky gave “a complete scheme-theoretic proof of Weil’s descent theorem” in an appendix to a paper on conjugation of Shimura varieties.

Some of us wondered at the time why Shimura had apparently omitted a crucial statement in his proof, although all the preliminary steps had been carefully prepared and were available for use by Milne and Varshavsky.  We decided that, in the small world of specialists to which Shimura belonged in the early 1960s, it would have been considered superfluous to mention the missing step.  When Shimura was consulted he claimed not to remember what he was thinking at the time.  In the (highly unlikely but not altogether inconceivable) event that number theory jettisons its current foundations in favor of Mochizuki’s inter-universal Teichmüller theory, collective memory loss may jeopardize preservation of present-day number theory even if it can be proved to be theoretically consistent with the new foundations.

Attribution:  The image on this page is from animations created by Etienne Farcot based on the specifications of Shinichi Mochizuki and suggestions of Ivan Fesenko. The image is licensed under a Creative Commons Attribution 4.0 International License.

2 thoughts on “abc and foundations

  1. Pingback: “Ideas are the currency of mathematics.” | Mathematics without Apologies, by Michael Harris

  2. Pingback: White smoke over Oxford? | Mathematics without Apologies, by Michael Harris

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